# Nonparametric estimation of scalar diffusions based on low frequency data is ill-posed

*Emmanuel Gobet*,
*Marc Hoffmann* () and
*Markus Reiß*

No 2002,57, SFB 373 Discussion Papers from Humboldt University of Berlin, Interdisciplinary Research Project 373: Quantification and Simulation of Economic Processes

**Abstract:**
We study the problem of estimating the coefficients of a diffusion (Xl, t 2:: 0); the estimation is based on discrete data Xn . . n = 0, 1, ... ,N. The sampling frequency delta t is constant , and asymptotics arc taken at the number of observations tends to infinity. We prove that the problem of estimating both the diffusion coefficient - the volatility - and the drift in a nonparametric setting is ill-posed: The minimax rates of convergence for Sobolev constraints and squared-crror lOBS coincide with that of a respectively first and second order linear inverse problem. To ensure ergodicity and limit technical difficulties we restrict ourselves to scalar diffusions living on a compact interval with reflecting boundary conditions. An important consequence of this result is that we can characterize quantitatively the difference between the estimation of a diffusion in the low frequency sampling case and inference problems in other related frameworks: nonparametric estimation of a diffusion based on continuous or high frequency data: but also parametric estimation for fixed delta, in which case root-N-consistent estimators usually exist. Our approach is based on the spectral analysis of the associated Markov semigroup. A rate-optimal estimation of the coefficients is obtained via the nonparametric estimation of an eigenvalue-eigenfunction pair of the transition operator of the discrete time Markov chain (X/lA, n = 0,1, ... ,N) in a suitable Sobolev norm: together with an estimation of its invariant density.

**Keywords:** Diffusion processes; Nonparametric estimation; Discrete Sarnpling; Low frequency data; Spectral approximation; Ill-posed problems (search for similar items in EconPapers)

**Date:** 2002

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**Persistent link:** https://EconPapers.repec.org/RePEc:zbw:sfb373:200257

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